For those who are in a pure math PhD program already, is there anything that you wish you did to better prepare for your current Phd studies?

I graduated from a small college, we didn't even offer things like topology. My closest contact with pure math is real analysis(1 semester), and abstract algebra(1 year). What books would you recommend me to read in the next 6 months? Are undergraduate textbooks enough, or should I try something like the graduate texts in mathematics?

I want to do number theory, but that can very well change after I enroll...is there anything that's "good for all" in the graduate pure math world?

I want to be able to do independent work sooner in grad school, but I feel my background so far only enables me to read and (struggle to) understand books/problems.

I remember asking one of my professors that question my senior year when I was asking which grad level classes are most beneficial. He said the one thing you can't learn too much of is linear algebra. It shows up everywhere. I'm assuming you've had the basic undergrad course, but it never hurts to really learn the theory behind it (vector spaces, dual spaces, tensors, etc.), something that may be covered either in advanced undergrad classes or beginning graduate classes.

Two other areas I was told to focus on being good at are real analysis and abstract algebra, since almost every graduate program tests those two subjects in the qualifying exams. If you're interested in number theory, studying complex analysis would also be useful I think, as well as some differential geometry and topology.

As far as level (and this is what I am personally doing), if you haven't had a solid background in the subject, start with an advanced undergrad / beginning graduate level text. For the subjects I mentioned, those would be (for example):

1) Linear Algebra: Linear Algebra and its Applications, Lax2) Abstract Algebra: Abstract Algebra, Dummit and Foote (this thing is a tome)3) Real Analysis: This really depends on how much you know. If you're comfortable with Rudin's Principles, try his Real and Complex Analysis.4) Complex Analysis: The Rudin book above, as well as Visual Complex Analysis (I love this one).5) Differential Geometry: This is a tough subject to break into I find. If your multivariable analysis is shaky, it might not be a bad idea to read up on those chapters in Rudin before moving on. Alternatively, Spivak's Calculus on Manifolds covers background material really well. After this, you can try Lee's Introduction to Smooth Manifolds to get to the nitty gritty of the subject.6) Topology: Munkre's text is the gold standard for beginners. For differential topology, Milnor's text is a classic.

If you have the opportunity, I would definitely recommend looking at a graduate-level textbook. I think the biggest difference between undergraduate math courses and graduate math courses is the fact that graduate math courses don't baby you as much. You're expected to do a fair amount of work to actually understand the material. That is, attending lectures and doing the homework might not be enough to fully understand the content of the course. You might have to work out various proofs yourself, do some independent reading, or work on exercises that you seek out yourself.

The same goes for textbooks. Most undergraduate textbooks explain everything in full detail. If you read a proof in an undergraduate textbook, (usually) all of the necessary information is there. You might have to work to fully understand the proof, but usually you don't need to take out a piece of paper and a pen to do so. In graduate textbooks it is often the case that proofs are "sketched" or left entirely to the reader. In either case, if you want to fully understand the proof you'll have to actually do some work to get to that point. Sometimes it's not actually necessary to do this. One piece of advice I've heard a number of times as a graduate student is this: "If you're very confident that you could prove a given result (if you wanted to), then you don't have to prove it. If you're not so sure, then you have to try and prove it."

I haven't taken number theory as a grad student so I don't have a recommendation for a number theory textbook. But, for something that's at least tangentially related, I'd recommend checking out "Algebra" by Larry Grove. It's a short book. Not too hard to read. But it does require the reader to do at least a little bit of work to understand everything completely.